Properties

Label 17.1.10.9a1.1
Base \(\Q_{17}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $F_{5}\times C_2$ (as 10T5)

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Error: Incorrect local algebra for Q(zeta3)

Defining polynomial

\(x^{10} + 17\) Copy content Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $10$
Ramification index $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{17}(\sqrt{17})$
Root number: $-1$
$\Aut(K/\Q_{17})$: $C_2$
This field is not Galois over $\Q_{17}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:undefined
Roots of unity:$16 = (17 - 1)$

Intermediate fields

$\Q_{17}(\sqrt{17})$, 17.1.5.4a1.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Canonical tower

Unramified subfield:$\Q_{17}$
Relative Eisenstein polynomial: \( x^{10} + 17 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{9} + 10 z^{8} + 11 z^{7} + z^{6} + 6 z^{5} + 14 z^{4} + 6 z^{3} + z^{2} + 11 z + 10$
Associated inertia:$4$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois degree: $40$
Galois group: $C_2\times F_5$ (as 10T5)
Inertia group: $C_{10}$ (as 10T1)
Wild inertia group: $C_1$
Galois unramified degree: $4$
Galois tame degree: $10$
Galois Artin slopes: $[\ ]$
Galois Swan slopes: $[]$
Galois mean slope: $0.9$
Galois splitting model:$x^{10} - 17$